Simulation of the Internal Wave of a Subsurface Vehicle in a Two-Layer Stratified Fluid

AKBARNEJAD BAIE Mohammad, PIROOZNIA Mahmood, and AKBARINASAB Mohammad

Simulation of the Internal Wave of a Subsurface Vehicle in a Two-Layer Stratified Fluid

AKBARNEJAD BAIE Mohammad1), PIROOZNIA Mahmood2), *, and AKBARINASAB Mohammad3)

1) Department of Physical Oceanography, Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran 2) Department of Geodesy, K. N. Toosi University of Technology, Tehran 1996715433,Iran 3) Department of Physical Oceanography, University of Mazandaran, Babol 4741613534, Iran

Internal waves are one of the various phenomena that occur at sea, and they affect acoustic equipment and sea density measurement equipment. In this study, internal waves are simulated using computational fluid dynamics method in the presence of a submarine in a pre-stratified fluid. Several scenarios were implemented by Froude number changes and submersible velocity by using the Navier-Stokesturbulence model. Results indicate that the realizableturbulence model gives better results than the RNGmodel and the internal waves flow in this model are well represented, which increases the wavelength of the internal waves by increasing the Froude number and floating velocity, while the internal angle of the Kelvin waves is decreased. We also observe that increasing the floating velocity causes the turbulent velocity contours to increase due to the drag coefficient and its relationship with the Reynolds number. The Reynolds number increases with the increasing velocity of the float motion. The results indicate the efficiency of this method in the discovery of subsurface objects.

internal waves; numerical simulation; computational fluid dynamics; turbulence models

1 Introduction

2 Procedure

This section studies the governing equations with boundary conditions and the choice of numerical algorithms for solving equations.

2.1 CFD

In this method, the numerical solution of these equations is provided by converting the differential equations governing some fluids into algebraic equations. Approximations of a linear equation system are obtained by dividing the desired region into smaller elements and applying boundary conditions for boundary nodes. Solving this system obtains the algebraic equations, velocity field, pressure, and temperature fields in the desired region. Using the results obtained from solving equations, the result of the forces acting on the surfaces, the drag and lift coefficients and the heat transfer coefficient could be calculated (Hoffmann and Chiang, 2004; Lanoye, 2006). Incompressible Navier-Stokes equations for slurry fluid and pressure-based solution have been used in cases of low fluid velocity (Newman, 1977). The pressure-based solution is a better function of convergence than density-based solubility in cases where the fluid is in the range of incompressible velocities.

In the model analysis of this study, the realizable K-ε model and the RNGmodel, which are two-equation turbulence models have been used due to the high Reynolds number of internal wave and being in the range of turbulence flows. Also, in the analysis of the problem, it was used initially as a steady-state solution, and then the transient solution was used to analyze the stable conditions of the flow around the submarine (Gil’fanov and Zaripov, 2009). In this study, the first-order upstream method was first used to discretize continuity and momentum equations. After ensuring convergence, due to the higher accuracy, the second-order upstream method was used to continue the solution and to interpolate the pressure because the fluid water has a relatively high density and volumetric forces in depth have significant effects. The weight-based method based on volumetric force was used, which obtains more acceptable results, as indicated by an investigation of various numerical methods.

The main problem in incompressible flows is the proper connection between the velocity and pressure fields. The SIMPLE method is based on the repetition in the correction of the pressure field in the relationship between pressure and velocity, thus making it more appropriate for use in simulations and to establish a connection between velocity and pressure. For most issues, the dimensionless error is an appropriate convergence criterion.Accordingly, the residuals should be less than a certain amount. In fluent, the process of solving equations will stop when the pressure convergence criterion, velocity, and turbulent kinetic energy are less than the user-defined value (in this study, 1e-6 is the convergence condition). We need to ensure that the residuals are decreasing, in which case we can conclude that the process of solving is converging.

2.1.1 Basic principal equation

In modeling the floating object movement, a volumetric approach to free surface modeling is used due to the complex shape changes of the free surface and the existence of the wave fracture phenomenon. By calculating the distribution of two-phase fluid in the entire computational range, we can assume that an effective fluid exists in the entire computational domain. Therefore, Navier-Stokes equations and governing equations for turbulence modeling are solved for an effective fluid with viscosity and volumetric mass. The volume ratio transfer equation is (Pandey and Sivasakthivel, 2011):

with Eq. (1) being solved, the volume ratio distribution, which is the percentage of the presence of two fluids within each computational cell, is given by Eq. (2)

The volumetric mass and viscosity of the effective fluid in each computational cell are calculated by relation (3)

In relation (3), subscripts 1 and 2 represent two fluid phases. In the following, the Navier-Stokes and continuity equations for an effective fluid are solved by calculating effective volumetric masses and effective velocity

In the above equation, there are the components of velocity, pressure, gravity force, and viscosity of turbulence. The value in Eq. (4) is due to the time average of the Navier-Stokes equation. As we know, the equilibrium equations for an incompressible stream with constant viscosity are given by Eq. (5)

This equation is valid for both laminar flow and turbulent flow. However, for turbulent flow, dependent variables such as time and pressure are all time dependent.

2.1.2 Momentum equations for turbulent flow

The momentum equations for an incompressible flow with constant viscosity are as follows:

The CFD model solves the corresponding equations on a real scale, thereby giving it an advantage over traditional physical modeling to ensure that the Reynolds and Froude similarities are satisfied (Rassaei, 2014).

2.1.3 Turbulence model (realizable)

The realizablemodel, which was proposed by Shih(1995), was used to simulate the turbulent flow. This model has a high degree of rotational motion and providesandnew equations that have more coordination with Reynolds values and also provides a new methodfor calculating numerical viscosity. The transmission equations for this model are as follows (Shih, 1995):

In the above relations,Gindicates the kinetic energy production of turbulenceis affected by the average speed.Galso indicates the production of kinetic energy of turbulenceaffected by buoyancy.σandσrepresent the turbulent Prandtl numbers ofand, respectively (Launder and Spalding, 1972).

2.1.4 Turbulence model (RNG)

The RNGmodel is obtained by using Navier-Stokes moment equations and assisted by the so-called renormalization group method. The transition equations of the RNGturbulent model are very similar to the transition equations of the standardmodel (Choudhury, 1993).

The difference between the RNGturbulent model and the standardturbulent model is the presence of a new term in thetransition equation, that is,. This term is expressed in terms of Eq. (10)

A notable point is that themethod is adopted in its RNG or realizable form according to previous work.

2.2 Numerical Simulation Processes with the Help of CFD

Here, the pre-processing, generating computational networks and the stratification of the simulation are described as bellow.

2.2.1 Geometrical simulation of floating object (pre-processing)

Following the information on the submarine geometry described below, the floating object geometry studied in the Ansys Fluent Design Modeler is simulated. After the floating object geometry was produced (Fig.1), the file was transferred to the software to perform mesh gridding. The dimensional information of the floating object (submarine) in the computational and realistic model is shown in Table 1.A SUBOFF submarine model is chosen for this research, which was designed by the Defense Advanced Research Projects Agency. The simulation region has a length and width of 522 and 210m, respectively, and the overall water depth is 260m.

Table 1 Geometric dimension of the floating object in the computational model

Fig.1 Isometric view from the front of the submarine.

2.2.2 Generating computational networks

The grid generation software used by Ansys Mesher software is used because of the complex geometry from an unstructured triangular grid. Irregular or unstructured networks have a high degree of accuracy in the boundary layer networking around the submarine body due to their high flexibility and good alignment.

Through the creation of discipline and the arrangement of the orthogonal network in several layers around the submarine wall, the boundary layer network causes local order and smaller network around the wall, thus increasing the accuracy of the resolution in the interference points. Hence, the shear stress on the submarine wall, which is an important component of the force of the drag movement, is calculated with high precision.

The mesh generation used in the model contains 3.5 million computational cells (Fig.2). The boundary conditions and the distance of the submarine model from different parts of the domain solution are selected as follows:

1) The flow input boundary has a constant and accelerated speed at a distance of 2.5 times the length of the submarine.

2) The flow outlet boundary with zero static pressure has a distance of 6 times the length of the submarine.

3) The body of the object with the condition of the wall is of the non-slip type.

4) The side walls of the domain solution exhibit the condition of free slip at a distance of 3 times the length of the submarine for uniform and accelerated motion in thedirection.

The assumed distances from the boundaries of the model are such that the submarine is far from colliding with the boundaries and the solution is independent of the computational domain (Lanoye, 2006).

After computational networks are generated, the boundary conditions and the method of solving are determined, and the independence of the computational grid is ensured to obtain an optimal computational grid. Discretization in Fluent software uses a volume control technique to convert the governing equations of the problem into algebraic equations. This technique involves integrating the governing equations into the control volume. Generally, theAnsys CFD solvers are based on the finite volume method. Then, the model is entered into Ansys Fluent solver to solve two-phase fluid equations. The data in this study were subjected to analyses both in static mesh and in the form of dynamic mesh layering (Fig.2). Dynamic mesh is used in this study. Thus, the time variations during submersion and floats are considered. This condition means that when static networks are used, only different behavioral patterns (parameters such as pressure) resulting from submarine movement in a special moment are observed. Commenting on the edits and creatable flows after the submarine movement is impossible; thus, this work does not provide an output of these flows. This study shows the eddies created after the submerged motion of the dynamic mesh. The wavelength of the inner waves in the longitudinal direction can be obtained from Eq. (13)

Therefore, the wavelength for a substructure float at 5kn velocity is about 4.23m, about 8.30m for a speed of 7kn, and about 24.39m for a speed of 12kn. To control the total number of network cells, we set the network scale to a length of 1m.

2.2.3 Stratification of the simulation environment and running different scenarios

In this physical simulation, the environment is stratified in two layers. The presence of strong layering is a necessary condition for the creation of small internal waves. For this purpose, the combination of two layers of fluid–namely, saltwater and freshwater–with different densities was used. The density of freshwater is 998kgm?3, and that of saltwater is 1085kgm?3. Saltwater is placed in the upper part of the computational domain due to the higher density in the lower computational realm and the lower water density. In simulations, with the fixed geometry figure described in the previous section, three different scenarios with submarine velocity variations are used, which are presented in Table 2. According to different scenarios and to perform simulations, the fluid input velocity to the realm is equal to 2.7, 3.5, and 6.6ms?1. The total simulated computational depth is 260m. Depending on the fact that the submarine is located at the margin of saltwater and freshwater at a depth of 130m, the analyses were conducted at a depth of 130m. In this study, depths of 130m for freshwater and saltwater are considered (coordinates (0–130) of freshwater and coordinates (130–260) of saltwater). In this study, the operation was performed by using a computer system with an Intel Core i7 processor and 16GB of RAM. The calculation time was about 500h for each case.

Table 2 Scenarios considered for submarine speed

3 Results and Discussion

In this section, analyzing the network and simulation of the internal in different scenarios are expressed.

3.1 Analyzing the Independence of the Network

In this work, unstructured networks are used to reduce the production time and network resolution. The main feature of this kind of meshing is that it can be easily implemented for complex geometries. The other advantage of unstructured networks is the ability to quickly implement network optimization. The important criterion for generating the network inside the boundary layer is y+, which refers to the dimensionless distance of the first node of the network relative to the surface. As a result of the flow fluid passage through the objects, vortices are formed behind them, thereby influencing the force on the body. For this purpose, the mesh is located on the back and in areas close to the smaller object than the other areas. Table 3 describes the mesh profile used to check the independence of the results from the mesh. Different methods are used to ensure the convergence of results, which can be considered for the parameter of the pressure coefficient or drag coefficient. In this work, the pressure coefficient is not suitable for determining the independence of the result, because the network size is affected by the viscosity parameter. Therefore, the drag coefficient quantity is used to select the optimal network. To compare the results in different networks, Table 3 calculates the amount of friction drag coefficient on the object at 2.57ms?1, and the results are presented in Fig.3.

As shown in the diagram, as the mesh shrinks, the amount of friction coefficient decreases; this process leads to a constant amount for increasingly small meshes. There- fore, the simulation of the meshing will be used to reduce the computation time and cost.

Table 3 Number and type of gridding

Fig.3 Drag coefficient variation in terms of network number.

3.2 Simulation Results in Different Scenarios

The simulation results were compared and evaluated in three different speed scenarios to present the results. In the following, the results in a selected scenario (floating speed equal to 6.17ms?1) is provided with the implementation of the realizabledisturbance model. The results of the disturbance model RNGare presented in the discussion section.

3.2.1 Comparing the results in different speed scenarios

The most important result of this simulation is the calculation of the drag force on the submarine body, because the engine must be able to produce such a force to move forward at the desired speed. Here 70% of the post-frictional force and 30% of the compression are used. The drag force enters the submarine according to Table 4.

Table 4 Drag force on the submarine

Figs.4 to 6 show the velocity of the fluid in the longitudinal plane passing through the center of the submarine.

A comparison of different speeds shows that as the submarine velocity increased, the range created by the submarine increased considerably at a speed of 6.17ms?1, and the end of the object is about 3 times the length of the submarine.

Fig.4 Contour of fluid velocity in the scenario of 6.17ms?1 with the implementation of the realizable k-ε turbulence model.

Fig.5 Fluid velocity contour in the 3/60ms?1 scenario by applying the realizable k-ε disturbance model.

Fig.6 Fluid velocity contour in the 2.57ms?1 scenario with the implementation of the realizable k-ε turbulence model.

The drag relationship with proportional motion speed is equivalent to the power of 2 (Eq. (14)). The simulation results in Table 4 also confirm this finding.

3.2.2 Simulation results in the selected scenario (6.17ms?1speed) by applying the realizabledisturbance model

The following figures show the results at a speed of 6.17ms?1. Fig.7 shows the submarine movement between two lower freshwater areas (blue areas) and saltwater area with a higher density (red areas).

Fig.8 shows the contour of the pressure distribution generated by the dynamic fluid passing through the submarine body, and the sum of these pressures forms the drag force. The location of most drag pressure is the stagnation point on the front of the submarine.

Fig.7 Underwater movement in a stratified fluid.

Fig.8 Fluid pressure contour on the submarine body at a speed of 6.17ms?1.

Fig.9 shows the velocity contour in the transverse shear planes of the submarine. This contour specifies the range of major submarine influences on the surrounding water flow as a sequence of disturbances. Combining this contour with a contour speed of longitudinal plane gives a full-length view of the range of submarine effects on the surrounding water fluid.

Figs.10 and 11 show the range of submarine effects on turbulence at the fluid velocity in the longitudinal planes.

The total drag force on a submarine at a speed of 6.17ms?1is 161375N, that is, to produce such a speed, the minimum engine should be able to create such a thrust. The relationship between the submarine speed and its effect on the parameters of activity of the internal waves are shown in Table 5. Figs.12–14 indicate the distribution of the internal height of the waves as a result of the submarine movement in the fluid at 2.57, 3.60, and 6.17ms?1, respectively, by applying the realizablemodel. Figs.15–17 show the distribution of the internal height of the waves as a result of submarine movement in the fluid at a velocity of 2.57, 3.60, and 6.17ms?1, respectively, by applying the RNGdisturbance model.

Fig.9 Fluid velocity contour in transverse planes around the submarine.

Fig.10 Fluid velocity contour in longitudinal planes around the submarine (isometric view).

Fig.11 Fluid velocity contour in longitudinal planes around a submarine (side view).

Table 5 Relationship between submarine speed and its effect on the parameters of the internal wave’s activity

3.3 Validation of the Results

In this study, the results are compared with the results of a similar work to validate their accuracy. Given the few resources available in this area, the closest study in terms of title and target is the work of Chang(2006), which is titled ‘Numerical simulation of internal waves due to submerged submarine in a two-layered stratified medium’ (Sutherland, 2006). The designed submerged sub- marine, which was then simulated, and the environmental conditions in which the submarine moves at different speeds are very close to the ones in our work (Table 6). In the following, RANS flow disturbances and the VOF method are used for the modeling of the current and the internal waves created by the submarine motion in water.

Fig.12 Distribution of the internal waves obtained from submarine motion in the fluid at a velocity of 2.57ms?1.

Fig.13 Distribution of the internal waves obtained from submarine motion in the fluid at a velocity of 3.60ms?1.

Fig.14 Distribution of the internal waves obtained from submarine motion in the fluid at a velocity of 6.17ms?1.

Fig.15 Same as Fig.12, but applying the RNG k-ε disturbance model.

Fig.16 Same as Fig.13, but applying the RNG k-ε disturbance model.

Fig.17 Same as Fig.14, but but applying the RNG k-ε disturbance model.

Table 6 Considered scenarios for submarine speeds in the study of Chang et al. (2006)

An examination of the results of the Chang(2006) shows that their results are consistent with ours.

1) In their work, the longitudinal wavelengths increase with the increase in the Froude length, whereas the Kelvin angle decreases, which correlates with the results obtained from this study. Table 4 of the present study shows the details.

2) Given the considered scenarios, at 2.57s, the Froude number is 0.068, which means that the surface waves of a stream are subcritical and the Kelvin angle is very small. Tables 2 and 5 of the present study show the details.

3) This study shows that, on the basis of the considered scenarios in two stratified layers, internal waves are created and exhibit a triple-lope pattern, which is similar to the motion of the ships.

Fig.18 shows that the wave height distribution in different scenarios, with increasing submarine velocity, increases in terms of range and scope (-axis). The results of this study and the analysis of turbulence parameters, such as turbulence velocity, are consistent. In this study, the turbulence velocity contours increased and the variations occurred in a greater range as the velocity increased.

Fig.18 Distribution of internal waves in different scenarios in the study of Chang et al. (2006) when the axis is Z=0.

4 Conclusions

In this study, the internal wave of a subsurface vehicle in a two-layer stratified fluid is simulated by considering different strategies and turbulence models. The calculations indicate that increasing the Froude number increases the wavelength of the internal waves and decreases the angle of the Kelvin waves (angle V decreases, see Figs.4 to 6 and 12 to 14). Given that the turbulence velocity contours are proportional to the Froude number and also the velocity of water turbulence increased with increasing floating velocity, we conclude that the Froude number increased with increasing floating speed.

In the subcritical flow, when=0.312, surface waves are Kelvin waves and the Kelvin angle is constant about 20?; these produced surface waves vary with the waves produced by ships. When the Froude number is between 0.01 and 0.1971, the produced waves are Kelvin waves, but their wavelength is very small.

The results show the relationship between submarine speed and its effect on the parameters of the internal waves. When the submarine velocity increased, the wavelength of the internal waves, the range of variations in the fluid turbulence velocity, and the range of variations of the fluid pressure increased.

The pressure distribution curve indicates that the location of the most stagnant points of drag pressure is in the front of the submarine.

A comparison of the realizabledisturbance model with the RNGdisturbance model in different scenarios shows that the realizablemodel is better than the RNGmodel, and the flow of internal waves in this model is well illustrated (Figs.12–17). A comparison of the pattern obtained from both models shows that the RNGmodel can reflect only part of the emerging eddies. However, the parts that can display the internal waves are fully matched by the pattern obtained from the realizablemodel. A comparison of the turbulence velocity contours obtained from these two models shows that the RNGdisturbance model exhibits fewer changes than the realizablemodel; this case is also true in relation to fluid pressure variations. Therefore, the realizableturbulence model is better than the RNGturbulence model for the modeling of subsurface wave tracking. The results show that the motion of the subsurface objects causes a change in fluid turbulence velocity and fluid pressure change. At present, hydrodynamic pressure detection sensors, which can compute fluid pressure variations, are used to detect these objects. In this study, to investigate the internal waves, 3–5 million cells were used to process the operation by using a computer with an Intel Core i7 processor and 16 GB of RAM. To achieve better results, we need to generate 20 million cells to simulate and verify small Froude numbers; this process requires powerful high-speed computers, such as computers with an Intel Core i7 processor and 128GB of RAM. The results show that numerical modeling based on CFD equations can simulate subsurface submarine waves moving in multilayer and stratified environments. A limitation of the CFD numerical method, which is based on viscosity theory, is that it can only simulate a substrate’s inflow field over a limited distance (a distance between about 15 and 25 submarines).

Acknowledgement

We would like to give special thanks to Prof. A. A. Bidokhti for his supports and helps. Also we would like to thank the reviewers for their comments that make the paper more interesting and informative.

Annotation

α: volume ratio distribution

:density

:viscosity

:pressure

:gravity acceleration

:the stress tensor

:eddy viscosity

μ: turbulent viscosity

:the coefficient of thermal expansion

C:the constant coefficient

:turbulent kinetic energy

:the rate of dissipation of turbulent kinetic energy

:the Blasius similarity variable

:velocity field

:kinematic energy

σand σ:turbulent Prantel numbers ofand, respectively

R: transition equation

Y:the fluctuating dilation in compressible turbulence

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. E-mail: ma.pirooznia@email.kntu.ac.ir

June 9, 2019;

March 26, 2020;

October 10, 2020

(Edited by Xie Jun)